## Set of 3 Pythagorean triangles equal perimeters and area in A.P.

Here are sets of 3 Pythagorean triangles with equal perimeters and area in arithmetic progression.

Note that the perimeters are multiples of 120.

The pattern breaks in this example:

The perimeter is not a multiple of 120.

Find other sets of 3 Pythagorean triangles with equal perimeters that are not multiples of 120.

………………..   #2   ………………..

multiples of 120 appear again :

Notice that the radius of incircle of the first triple in each set is a multiple of 120

………………..   #3   ………………..

No multiples of 120 :

………………..   #4   ………………..

One triple is primitive in each set:

Find other such sets – that include a primitive – as shown in #4

math grad - Interest: Number theory
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### 2 Responses to Set of 3 Pythagorean triangles equal perimeters and area in A.P.

1. Paul says:

Here are a few sets

{315,624,699,1638}
{364,585,689,1638}
{455,504,679,1638}
Areas are 98280 & 106470 & 114660 AP is 8190

{3150,6240,6990,16380}
{3640,5850,6890,16380}
{4550,5040,6790,16380}
Areas are 9828000 & 10647000 & 11466000 AP is 819000

{3465,6864,7689,18018}
{4004,6435,7579,18018}
{5005,5544,7469,18018}
Areas are 11891880 & 12882870 & 13873860 AP is 990990

{630,1248,1398,3276}
{728,1170,1378,3276}
{910,1008,1358,3276}
Areas are 393120 & 425880 & 458640 AP is 32760

{1575,3120,3495,8190}
{1820,2925,3445,8190}
{2275,2520,3395,8190}
Areas are 2457000 & 2661750 & 2866500 AP is 204750

{2730,10472,10822,24024}
{3575,9912,10537,24024}
{4620,9152,10252,24024}
Areas are 14294280 & 17717700 & 21141120 AP is 3423420

{945,1872,2097,4914}
{1092,1755,2067,4914}
{1365,1512,2037,4914}
Areas are 884520 & 958230 & 1031940 AP is 73710

{1890,3744,4194,9828}
{2184,3510,4134,9828}
{2730,3024,4074,9828}
Areas are 3538080 & 3832920 & 4127760 AP is 294840

{3825,8004,8871,20700}
{4500,7475,8725,20700}
{5796,6325,8579,20700}
Areas are 15307650 & 16818750 & 18329850 AP is 1511100

{1260,2496,2796,6552}
{1456,2340,2756,6552}
{1820,2016,2716,6552}
Areas are 1572480 & 1703520 & 1834560 AP is 131040

{5355,10608,11883,27846}
{6188,9945,11713,27846}
{7735,8568,11543,27846}
Areas are 28402920 & 30769830 & 33136740 AP is 2366910

{4590,6496,7954,19040}
{4879,6240,7921,19040}
{5440,5712,7888,19040}
Areas are 14908320 & 15222480 & 15536640 AP is 314160

{5460,20944,21644,48048}
{7150,19824,21074,48048}
{9240,18304,20504,48048}
Areas are 57177120 & 70870800 & 84564480 AP is 13693680

{3780,7488,8388,19656}
{4368,7020,8268,19656}
{5460,6048,8148,19656}
Areas are 14152320 & 15331680 & 16511040 AP is 1179360

{4725,9360,10485,24570}
{5460,8775,10335,24570}
{6825,7560,10185,24570}
Areas are 22113000 & 23955750 & 25798500 AP is 1842750

{2205,4368,4893,11466}
{2548,4095,4823,11466}
{3185,3528,4753,11466}
Areas are 4815720 & 5217030 & 5618340 AP is 401310

{2520,4992,5592,13104}
{2912,4680,5512,13104}
{3640,4032,5432,13104}
Areas are 6289920 & 6814080 & 7338240 AP is 524160

{4485,5980,7475,17940}
{4715,5772,7453,17940}
{5244,5265,7431,17940}
Areas are 13410150 & 13607490 & 13804830 AP is 197340

{2835,5616,6291,14742}
{3276,5265,6201,14742}
{4095,4536,6111,14742}
Areas are 7960680 & 8624070 & 9287460 AP is 663390

{5670,11232,12582,29484}
{6552,10530,12402,29484}
{8190,9072,12222,29484}
Areas are 31842720 & 34496280 & 37149840 AP is 2653560

{5040,9984,11184,26208}
{5824,9360,11024,26208}
{7280,8064,10864,26208}
Areas are 25159680 & 27256320 & 29352960 AP is 2096640

{4410,8736,9786,22932}
{5096,8190,9646,22932}
{6370,7056,9506,22932}
Areas are 19262880 & 20868120 & 22473360 AP is 1605240

{4095,8112,9087,21294}
{4732,7605,8957,21294}
{5915,6552,8827,21294}
Areas are 16609320 & 17993430 & 19377540 AP is 1384110

{5985,11856,13281,31122}
{6916,11115,13091,31122}
{8645,9576,12901,31122}
Areas are 35479080 & 38435670 & 41392260 AP is 2956590

{6615,13104,14679,34398}
{7644,12285,14469,34398}
{9555,10584,14259,34398}
Areas are 43341480 & 46953270 & 50565060 AP is 3611790

{3927,21060,21423,46410}
{5005,20400,21005,46410}
{6188,19635,20587,46410}
Areas are 41351310 & 51051000 & 60750690 AP is 9699690

Paul.

• benvitalis says:

I’ve rearranged some of your results in 3 parts.
Find other sets that include a primitive triple, as shown in #4